## 7.5. Methods of determining scale of a vertical photograph

Scale from photo and ground measurements

Various methods are used to determine the scale of a photograph, depending on the types of information available. The basic and most straightforward technique of expressing scale is the one that uses the ratio of the distance between two points on a photograph (PD) to the distance between the same two points on the ground (GD). From the basic geometry of a vertical aerial photograph (figure 7.6 ), we can see that from similar triangles Lop and LOP, photo scale may be expressed as:

Since

Therefore,

Similarly, map scale may be expressed as:

where:

PS is the photo scale,
MS is the map scale,
PD is the photo distance measured between two well identified points on the photograph,
MD is the map distance measured between two points on the map, and
GD is the ground distance between the same two points on the photograph (or on the map), expressed in the same units.

As mentioned earlier, the photo scale reciprocal (PSR), sometimes called photo scale factor or the map scale reciprocal (MSR), sometimes called map scale factor is simply the inverse of the photo scale or the map scale and can be expressed as:

Example 7.4

Assume that the distance between two points was measured to be 83.33 mm on a vertical photograph and 125.00 mm on a map. If the surveying ground distance between the same two points is 3000 m, what are the scales and the scale reciprocals of the photograph and the map?

Solution:

From equations (7.4) and (7.5), we obtain:

and

and from equations (7.6) and (7.7), we obtain:

Example 7.5

Using the information in example 7.4 express the photo scale and the map scale in unit equivalents.

solution:

and

Equations 7.4 and 7.5 are convenient when the distance between any well identified points on a photograph -or a map- is accurately measured and their corresponding ground distance is known. Unfortunately, many objects are not always well identified on a photograph and surveying ground distances may not always be available. In addition, accurate measurements of these ground distances are tedious, time consuming, and costly, particularly for long distances and/or when a higher number of measures are needed for accurate average project scale determination. This situation suggests that other alternatives must be considered to determine the photo scale.

Scale from focal length and flying height

Generally, a photograph is annotated with information, including the date and time of acquisition, the project code, the serial identification of the photograph (i.e., line number and exposure number), the focal length, the flying altitude of the aircraft above MSL or above datum (figure 7.7).

When these parameters are present, the photo scale may be determined using the same geometry of a vertical aerial photograph in figure 7.6. Again, from similar triangle Lop and LOP, we may write:

where:

PS is the photo scale,
f is the focal length of the camera used to take the photograph, and
H-h is the flying altitude of the aircraft above the ground, which may be the average elevation of the entire project area, the average elevation of the photograph -or between two points on the photograph, or the elevation of a single point on the ground.

This technique is fairly accurate, particularly in determining point scale, when the flying height of the aircraft above MSL and the ground elevation are precisely known. For average scales, the accuracy of this technique depends mainly on the terrain. In addition, with the use of a conventional altimeter on the aircraft, flying height above the ground may not be recoverable. Further, in rugged terrain, the flying height values may not represent an accurate value for the project area, the photograph, or even a point on the photograph, since the flying height printed on the photographs is only a nominal (usually the average) value for the entire project. This is why the photo -or project- scale is usually referred to as nominal rather than true scale. Only maps are supposed to have a true scale and they can be very useful in determining photo scale as well.

Scale from photo and map measurements

If the information on the photograph and the ground distance between two points are not available, the photo scale can still be determined if a map of the area is available. By examining equations 7.6 and 7.7, it can be inferred that:

GD = PD ´ PSR

and

GD = MD ´ MSR

It follows that:

PR ´ PSR = MD ´ MSP

and

PD and MD are distances measured between the same 2 points that are well identified on the photograph and the map, respectively (figure 7.8). For example, in figure 7.8, if the distance between the same two points, A and B, were measured to be 6.05 cm on a photograph and 4.25 cm on a 7.5-minute USGS quadrangle (PRS = 24,000), then the scale of the photograph (PS) would be 6.05 ¸ (4.25´24000) = 1:16860.

This method is the most useful and the most practical form of determining a photo scale, simply because of easily obtainable values from a map. A map will always have a scale value or, at least, a bar scale from which a scale value can be determined. Further, distances between two points can easily and more accurately be measured on a map.

However, many countries and many regions are not entirely covered with maps or ground surveying distances. If a situation arises where ground surveying distances, a map of the area, and the information on a photograph are not available, a photo scale can still be determined if another photograph of the same area and with known scale is available.

Scale from an existing aerial photograph

This situation requires that either the scale or the focal length and the flying height above the ground of a second photograph are known. When the scale of a second photograph (PS2) is known, our photo scale (PS1) may be determined in the same manner as it was done using a map. By measuring the distances between the same two points on photo 1 (PD1) and photo 2 (PD2), the unknown scale of photo 1 (PS1) may be determined from the known scale of photo 2 (PS2) as:

PS1 is the scale to be determined for the photograph considered (photo 1),
PD1 is the distance measured between two well identified points on photo 1,
PD2 is the distance measured between the same two points on the known-scale photograph (photo 2), and
PSR2 is the scale reciprocal of photo 2.

If the scale of photo 2 is not known but the focal length and the flying height above the ground are available, then, the scale of photo 1 may be determined as follows:

From equation 7.8 we can write

Substituting equation 7.11 in equation 7.10, we obtain:

where:

f2 is the focal length of the camera used to take photo 2 and
H-h2 is the flying height of the aircraft above the ground of photo 2.

Procedure of determining scale for a single photograph

If the information usually printed on an aerial photograph is missing but a map of the area or ground measurements are available, the average scale of a single photograph may be determined as follows:

1. Select two points, preferably at the average ground elevation of the photograph, and approximately equidistant from and opposite the photo center. The two points must accurately be located on a map -or on the ground if a map is not available.
2. Accurately (to nearest 0.1 mm) measure the distance between the two points on the photograph and the map -or determine the ground horizontal distance (to nearest 1 m) if a map is not available.
3. Compute the photo scale using equation 8.9 if map measurements are available or using equation 8.4 if ground measurements are available.
Example 7.6
Two points were clearly identified on an aerial photograph and on a 1:24000-scale USGS topographic map of the same area. The distance on the photograph between the two points was measured to be 8.25 cm and the distance on the map between the same points was measured to be 5.72 cm. Find the scale of the photograph.

Solution:

By equation 7.9, we obtain:

If the information on the photograph is available, then the photo scale is determined using equation 7.8.
Example 7.7
Suppose a photograph was taken from 3500 m above MSL with a 152.4-mm focal length camera as printed on the photograph. If the average ground elevation of the area covered by the photograph is 830 m above MSL, what is the scale of the photograph?

Solution:

Using equation 7.8,

Example 7.8

Now, suppose that neither a map nor the information on the photograph are available, but the ground distance between two points well identified points is known to be 1320 m and the distance measured on the photograph between the same two points is measured to be 7.86 cm. Find the scale of the photograph.

Solution:

By equation 7.4, we obtain,

Procedure of determining scale for a series of photographs

The average scale for the entire project or a series of photographs may be determined as follows:

1. Determine the average elevation for each flight line using a topographic map.
2. Compute the average altitude for the area considered using the average altitudes determined in step 1 and apply equation 7.8.
3. Apply equation 7.2 if H and f are known, otherwise, use the same steps as in the case of a single photograph by considering the average elevation of a set of photograph instead of one single photograph.

4.
Example 7.9

A 305-mm focal length was used to take aerial photographs from 4000 m above MSL. Using a topographic map, the average elevations of the flight lines were found to be 700 m for line 1, 580 m for line 2, 650 m for line 3, and 750 for line 4. Find the average scale of the project.

Solution:

First, the average elevation of the entire area is determined as:

Then, using equation 8.2, we obtain the average scale of the project as:

Example 7.10

In example 7.9, the information (H and f) on the photographs was missing, but a number of surveying benchmarks were identified on the photographs and a topographic map of the same area. Two benchmark points were identified on line 1 and line 4 and their coordinates were found to be XA = 515,000 m; YA = 5,182,000 m, XB = 518,000 m; and YB = 5,179,000 m. By properly overlapping (mosaiquing) the photographs and measuring the photo distance between these two points, it was found to be 26.5 cm. Find the average scale of the project.

Solution:

To be able to find the photo scale, we first need to determine the ground distance between the two points. Since we know the ground coordinates, the distance between the two points may be determined using the Pythagorean theorem:

Then, using equation 7.4, the photo scale can be determined as:

Another alternative is to compare known areas on the ground to their imaged areas on the photograph. The area ratio is directly proportional to the square of the photo scale. This latter may be determined as follows:

PS = PD / GD

therefore, PS2 = PD2 / GD2

It follows that:

and

Example 7.11

A rectangular corn field was identified on an aerial photograph and its length and width were measured to be 10 cm and 6.5 cm, respectively. The ground area of the field is known to be 4 hectares. Find the scale of the photograph.

Solution:

The area of the field on the photograph is computed as:

Photo area = 10 cm ´ 6.5 cm = 65 cm2

Then, using equation 7.13, we obtain:

Remember that, in order to find PS, both photo area and ground area must be expressed in the same units. There are 100,000,000 cm2 in a hectare (or 6,272,640 in2 in an acre and 2.471 acres in a hectares).

7.4 Types of Scale | 7.5 Methods of Determining Scale of a Vertical Photograph | 7.6  Determining Scale of a Tilted Photograph